Sequential loser-elimination method

A sequential loser-elimination method is a method that works by repeatedly eliminating the loser of another voting method until a single candidate remains, and then electing that candidate. The method that is used to determine the loser is called the base method.

Instant runoff voting (without whole-votes-equal ranking) is a sequential loser-elimination method based on First past the post, and Baldwin is a sequential loser-elimination method based on the Borda count.

Proportional Systems

Within the class of multi-member systems there are Sequential Systems and Optimal Systems. The Sequential Elimination systems make up one of the two types of Sequential Systems. The other type is Sequential Selection systems and they tend to be Cardinal voting systems.

Criteria

It is a common misconseption that it is the Ordinal Ballot which causes such systems to fail monotonicity. It is in fact due to the Sequential loser-elimination method. The confusion comes from the fact that one can only do sequential elimination with ordinal ballots not sequential selection. It can be easialy seen that most Cardinal sequential elimination methods are nonmonotonic.

Proving criterion compliances for loser-elimination methods often use inductive proofs, and can thus be easier than proving such compliances for other method types. For instance, if the base method passes the majority criterion, a sequential loser-elimination method based on it will pass mutual majority. Loser-elimination methods are also not much harder to explain than their base methods. However, loser-elimination methods often fail monotonicity due to chaotic effects (sensitivity to initial conditions).

When the base method passes local independence of irrelevant alternatives, the loser-elimination method is equivalent to the base method.

If the base method satisfies a criterion for a single candidate (e.g. the majority or Condorcet criterion), then a sequential loser-elimination method satisfies the corresponding set criterion (e.g. mutual majority or Smith) if eliminating a candidate can't remove another candidate from the set in question. This because when all but one of the candidates of the set have been eliminated, the single-candidate criterion applies to the remaining candidate.

If one base method is used as a tiebreaker for another, and both base methods pass the candidate criterion, then the sequential loser-elimination method satisfies the set criterion. If only one of them passes the candidate criterion, then the elimination method need not pass the set criterion.

Notes

Note that IRV with whole-votes equal-ranking may not be a sequential-loser elimination method depending on which rules are used to determine the winner; see the STV#Ways of dealing with equal rankings section.

Note that though a voting method may be a sequential loser-elimination method in its single-winner case, it may not be so under certain generalizations of the criterion to the multi-winner case. Consider the following 2-winner example for STV with Droop quotas:

 99 A>B  
 1 C  

A and B would win. However, if the criterion for a multi-winner sequential loser-elimination method is that it must repeatedly eliminate until only (# of winners) candidates remain, with no surplus distribution being done, and with those remaining candidates winning, then A and C would win, since B is the candidate with the fewest 1st choices here. So for the multi-winner or proportional case, it may be required to allow surplus distribution or other steps in order to best generalize the sequential loser-elimination criterion for the multi-winner case.